François Jouve

A level-set method for shape optimization

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh. To cite this article: G. Allaire et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1125–1130.

Damage and fracture evolution in brittle materials by shape optimization methods

This paper is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.

Compact Unstructured Representations for Evolutionary Design

This paper proposes a few steps to escape structured extensive representations for objects, in the context of evolutionary Topological Optimum Design (TOD) problems: early results have demonstrated the potential power of Evolutionary methods to find numerical solutions to yet unsolved TOD problems, but those approaches were limited because the complexity of the representation was that of a fixed underlying mesh. Different compact unstructured representations are introduced, the complexity of which is self-adaptive, i.e. is evolved by the algorithm itself. The Voronoi-based representations are variable length lists of alleles that are directly decoded into object shapes, while the IFS representation, based on fractal theory, involves a much more complex morphogenetic process. First results demonstrates that Voronoi-based representations allow one to push further the limits of Evolutionary Topological Optimum Design by actually removing the correlation between the complexity of the representations and that of the discretization. Further comparative results among all these representations on simple test problems seem to indicate that the complex causality in the IFS representation disfavors it compared to the Voronoi-based representations.

Eigenfrequency optimization in optimal design

We maximize the first eigenfrequency, or a sum of the first ones, of a bounded domain occupied by two elastic materials with a volume constraint for the most rigid one. A relaxed formulation of this problem is introduced, which allows for composite materials as admissible designs. These composites are obtained by homogenization of fine mixtures of the two original materials. We prove a saddle-point theorem that permits to reduce the full (unknown) set of admissible composite designs to the smaller set of sequential laminates which is explicitly known. Although our relaxation theorem is valid only for two non-degenerate materials, we deduce from it a numerical algorithm for eigenfrequency optimization in the context of optimal shape design (i.e. when one of the two materials is void). As is the case with all homogenization methods, our algorithm can be seen as a topology optimizer. Numerical results are presented for various two- and three-dimensional problems.

Computer Simulation of Arcuate Keratotomy for Astigmatism

BACKGROUND The development of refractive corneal surgery involves numerous attempts to isolate the effect of individual factors on surgical outcome. Computer simulation of refractive keratotomy allows the surgeon to alter variables of the technique and to isolate the effect of specific factors independent of other factors, something that cannot easily be done in any of the currently available experimental models. METHODS We used the finite element numerical method to construct a mathematical model of the eye. The model analyzed stress-strain relationships in the normal corneoscleral shell and after astigmatic surgery. The model made the following assumptions: an axisymmetric eye, an idealized aspheric anterior corneal surface, transversal isotropy of the cornea, nonlinear strain tensor for large displacements, and near incompressibility of the corneoscleral shell. The eye was assumed to be fixed at the level of the optic nerve. The model described the acute elastic response of the eye to corneal surgery. RESULTS We analyzed the effect of paired transverse arcuate corneal incisions for the correction of astigmatism. We evaluated the following incision variables and their effect on change in curvature of the incised and unincised meridians: length (longer, more steepening of unincised meridian), distance from the center of the cornea (farther, less flattening of incised meridian), depth (deeper, more effect), and the initial amount of astigmatism (small effect). CONCLUSIONS Our finite element computer model gives reasonably accurate information about the relative effects of different surgical variables, and demonstrates the feasibility of using nonlinear, anisotropic assumptions in the construction of such a computer model. Comparison of these computer-generated results to clinically achieved results may help refine the computer model.

Coupling the Level Set Method and the Topological Gradient in Structural Optimization

A numerical coupling of two recent methods in shape and topology optimization of structures is proposed. On the one hand, the level set method, based on the shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes. On the other hand, the bubble or topological gradient method is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields an efficient algorithm which can escape from local minima. It have a low CPU cost since it captures a shape on a fixed Eulerian mesh. The main advantage of our coupled algorithm is to make the resulting optimal design more independent of the initial guess.

Optimal design of micro-mechanisms by the homogenization method

The design of mechanisms for building micro-tools can be viewed as a shape optimization problem with a peculiar objective function. We propose such an optimization method based on homogenization, which is called topology optimization.

The homogenization method for topology and shape optimization. Single and multiple loads case

ABSTRACT This paper is devoted to an elementary introduction to the homogenization methods applied to topology and shape optimization of elastic structures under single and multiple external loads. The single load case, in the context of minimum compliance and weight design of elastic structures, has been fully described in its theoretical as well as its numerical aspects in [4]. It is here briefly recalled. In the more realistic context of “multiple loads”, i.e. when the structure is optimized with respect to more than one set of external forces, most of the obtained theoretical results remain true. However, the parameters that define optimal composite materials cannot be computed explicity. In this paper, a method to treat numerically the multiple loads case is proposed.

A level set method for the numerical simulation of damage evolution

The aim of this article is to present an introduction to dissipation inequalities and to present some well known and some recent results in this area. Mathematics Subject Classification (2000). Primary 93C10; Secondary 93D99.The first part of this article concerns visibility, that is the question of determining the internal properties of a medium by making electromagnetic measurements at the boundary of the medium. We concentrate on the problem of Electrical Impedance Tomography (EIT) which consists in determining the electrical conductivity of a medium by making voltage and current measurements at the boundary. We describe the use of complex geometrical optics solutions in EIT. In the second part of this article we will review recent theoretical and experimental progress on making objects invisible to electromagnetic waves. This is joint work with A. Greenleaf, Y. Kurylev and M. Lassas. Maxwell’s equations have transformation laws that allow for design of electromagnetic parameters that would steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was hidden. The object would have no shadow. New advances in metamaterials have given some experimental evidence that this indeed can be made possible at certain frequencies. Mathematics Subject Classification (2000). Primary 35R30, 78A46 ; Secondary 58J05, 78A10 .Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction in order to significantly progress understanding. This article is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. In this context there is an array of problems with a common mathematical structure, namely that the probability measure in question is a change of measure from a Gaussian. We illustrate the wide-ranging applicability of this structure. For problems whose solution is determined by a probability measure on function space, information about the solution can be obtained by sampling from this probability measure. One way to do this is through the use of Markov chain Monte-Carlo (MCMC) methods. We show how the common mathematical structure of the aforementioned problems can be exploited in the design of effective MCMC methods.

Structural Optimization by the Level-Set Method

In the context of structural optimization, we describe a new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation. We have implemented this method in two and three space dimensions for models of linear or non-linear elasticity, with various objective functions and constraints on the volume or on the perimeter. The shape derivative is computed by an adjoint method. The cost of our numerical algorithm is moderate since the shape is captured on a fixed Eulerian mesh. Although this method is not specifically designed for topology optimization, it can easily handle topology changes.